By Agence France-Presse
While the historic gathering in Singapore is the big story of the week, key central bank decisions in the US and Europe will also be closely followed.
Trump pulled out of endorsing a joint communique after the G7 meet finished on Saturday in a row over trade, accusing the summit's chairman, Canadian Prime Minister Justin Trudeau, of dishonesty.
The shock decision followed a testy gathering in Quebec, where the US president came under fire for his "America First" protectionist drive that has fuelled fears of a global trade war.
He had left early for Singapore, only to take exception to comments by Trudeau at a news conference.
"One thing that we do know for sure is the president's uncontrollable need to defend his status is more apparent than any strategy when (it) comes to bilateral trade negotiations," said Stephen Innes, head of Asia-Pacific trade at OANDA.
"But, the far from harmonious Quebec summit confirmed deep-seated G6+1's expanding policy fissures on a plethora of significant concerns including climate change, the Iran nuclear deal and, of course, trade."
"While expectations were not exactly high going into the meeting, the result was a bit worse than even the markets' dismal presuppositions."
Fed, ECB in focus
Tokyo ended the morning session 0.3 percent up, Hong Kong added 0.2 percent and Seoul also gained 0.3 percent.
Shanghai dipped 0.3 percent and Singapore was slightly lower, while Wellington, Taipei and Manila were also down.
Sydney was closed for a public holiday.
All eyes are now on Singapore, with the leaders of North Korea and the United States due to meet for the first time in history, with Pyongyang's nuclear programme top of the agenda.
Tai Hui, JP Morgan Asset Management chief market strategist for Asia-Pacific, said in a note: "Although the... summit in Singapore is capturing public attention, the direct impact on markets is likely to be limited."
"Ongoing dialogue between Washington and Pyongyang is positive, since this implies military action is on hold, limiting any economic implication for South Korea and surrounding region. Investors will hope to see steady progress with opportunities for more discussion in coming months."
This week is also expected to see the Federal Reserve lift interest rates again, and traders will be poring over its post-meeting statement for an idea about future moves.
That will be followed by the European Central Bank's latest gathering, which could see policymakers discuss winding down its crisis-era stimulus programme.
Key figures around 0230 GMT
Tokyo - Nikkei 225: UP 0.3 percent at 22,762.94 (break)
Hong Kong - Hang Seng: UP 0.2 percent at 31027.09
Shanghai - Composite: DOWN 0.3 percent at 3,057.26
Euro/dollar: DOWN at $1.1798 from $1.1800 at 2100 GMT on Friday
Pound/dollar: UP at $1.3417 from $1.3400
Dollar/yen: UP at 109.50 yen from 109.47 yen
Oil - West Texas Intermediate: DOWN seven cents $65.67 per barrel
Oil - Brent Crude: UP 11 cents at $76.57 per barrel
New York - Dow Jones: UP 0.3 percent at 25,316.53 (close)
London - FTSE 100: DOWN 0.3 percent at 7,681.07 (close)